Solving Mixed Fraction Division $5 \frac{2}{3} \div \frac{1}{4}$ A Step-by-Step Guide

Introduction

In the realm of mathematics, understanding fractions and their operations is pivotal. This article delves into the intricacies of dividing mixed fractions, specifically focusing on the expression $5 \frac{2}{3} \div \frac{1}{4}$. We aim to not only solve this problem but also to provide a comprehensive understanding of the steps involved, ensuring clarity and confidence in handling similar mathematical challenges. By breaking down each step and elucidating the underlying principles, this guide serves as a valuable resource for students, educators, and anyone keen on honing their mathematical skills.

Problem Statement: $5 \frac{2}{3} \div \frac{1}{4}$

The problem at hand involves the division of a mixed fraction by a simple fraction. Mixed fractions, which combine a whole number and a fraction, often pose a challenge if not handled correctly. Dividing by a fraction is another concept that requires a firm grasp of mathematical rules. Before diving into the solution, it’s crucial to understand these concepts thoroughly. We will begin by converting the mixed fraction into an improper fraction, which simplifies the division process. This transformation is a fundamental step in solving such problems and lays the groundwork for the subsequent operations. Understanding the nature of fractions and how they interact is essential for mastering arithmetic and algebra. Therefore, this article will meticulously explain each step, ensuring the reader comprehends not only the method but also the logic behind it. By the end of this guide, you should be equipped to tackle similar problems with ease and accuracy. Remember, practice is key, and consistent application of these principles will solidify your understanding and skills in fraction arithmetic.

Step 1: Converting the Mixed Fraction to an Improper Fraction

The initial step in solving $5 \frac{2}{3} \div \frac{1}{4}$ is to convert the mixed fraction $5 \frac{2}{3}$ into an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion is necessary because it simplifies the division process significantly. To convert a mixed fraction to an improper fraction, you multiply the whole number part by the denominator of the fractional part and then add the numerator. The result becomes the new numerator, and the denominator remains the same. In our case, we multiply 5 (the whole number) by 3 (the denominator) to get 15. Then, we add 2 (the numerator) to get 17. So, the new numerator is 17, and the denominator remains 3. Therefore, the improper fraction equivalent to $5 \frac{2}{3}$ is $\frac{17}{3}$. This conversion is a crucial step because it transforms the mixed fraction into a form that is easier to manipulate in division. Understanding this process is fundamental to working with fractions and is a skill that is applied across various mathematical contexts. By mastering this conversion, you’ll find fraction operations much more manageable, paving the way for more complex problem-solving.

Step 2: Dividing Fractions - Multiplying by the Reciprocal

Now that we've converted the mixed fraction $5 \frac2}{3}$ to the improper fraction $\frac{17}{3}$, we can proceed with the division. The original problem is now reframed as $\frac{17}{3} \div \frac{1}{4}$. The key to dividing fractions is to understand that division is equivalent to multiplying by the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. In this case, the divisor is $\frac{1}{4}$, so its reciprocal is $\frac{4}{1}$, which is simply 4. Therefore, dividing by $\frac{1}{4}$ is the same as multiplying by 4. Our problem now transforms into a multiplication problem $\frac{17{3} \times 4$. This conversion from division to multiplication by the reciprocal is a fundamental rule in fraction arithmetic. It simplifies the process and allows us to apply the more straightforward rules of multiplication. Understanding this principle not only helps in solving this specific problem but also provides a valuable tool for handling any fraction division. By mastering this technique, you’ll find fraction operations more intuitive and less daunting, which is crucial for building a strong foundation in mathematics.

Step 3: Performing the Multiplication

Having transformed the division problem $\frac{17}{3} \div \frac{1}{4}$ into the multiplication problem $\frac{17}{3} \times 4$, the next step is to perform the multiplication. When multiplying a fraction by a whole number, you multiply the numerator of the fraction by the whole number, while the denominator remains the same. In this case, we multiply 17 (the numerator of $\frac{17}{3}$) by 4 (the whole number) to get 68. The denominator, 3, remains unchanged. So, the result of the multiplication is $\frac{68}{3}$. This fraction, $\frac{68}{3}$, is an improper fraction, meaning the numerator is greater than the denominator. While this is a correct answer, it's often more useful to express it as a mixed number to better understand its magnitude. The multiplication step is straightforward, but it's crucial to ensure accurate arithmetic to arrive at the correct answer. This step highlights the importance of understanding basic multiplication rules and how they apply to fractions. By mastering this simple multiplication, you’re one step closer to confidently handling more complex fraction problems. The ability to accurately multiply fractions is a foundational skill that is essential in various areas of mathematics.

Step 4: Converting the Improper Fraction to a Mixed Number

After performing the multiplication, we arrived at the improper fraction $\frac{68}{3}$. To better understand the value of this fraction, we need to convert it back into a mixed number. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). To convert an improper fraction to a mixed number, you divide the numerator by the denominator. The quotient (the result of the division) becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator remains the same. In our case, we divide 68 by 3. The quotient is 22, and the remainder is 2. Therefore, the whole number part of our mixed number is 22, and the fractional part is $\frac{2}{3}$. So, the mixed number equivalent to $\frac{68}{3}$ is $22 \frac{2}{3}$. This conversion is important because it provides a clearer sense of the fraction’s value. A mixed number gives a more intuitive understanding of the quantity being represented. Converting improper fractions to mixed numbers is a valuable skill in various mathematical contexts, and it’s particularly useful when dealing with real-world applications where understanding the magnitude of a number is crucial. By mastering this conversion, you enhance your ability to interpret and work with fractions effectively.

Step 5: Determining the Range

Now that we have the result $22 \frac{2}{3}$, the final step is to determine the range within which this number falls. The question asks for the pair of numbers between which the value of c lies. We found that $c = 22 \frac{2}{3}$, which is a number greater than 22 but less than 23. Therefore, the value of c lies between 22 and 23. This step requires an understanding of number ranges and the ability to compare numbers, including mixed numbers. By converting the improper fraction to a mixed number, we made it easier to visualize and place the value within the correct range. Understanding number ranges is a fundamental skill in mathematics and is essential for estimating and approximating values. It also helps in checking the reasonableness of answers. In this case, we were able to confidently determine that $22 \frac{2}{3}$ falls between 22 and 23, which aligns with our calculations and understanding of fractions. This final step reinforces the importance of a holistic understanding of numbers and their properties.

Conclusion

In conclusion, we have successfully solved the problem $5 \frac{2}{3} \div \frac{1}{4}$, and determined that the value of c is $22 \frac{2}{3}$, which lies between 22 and 23. This exercise involved several key steps, including converting a mixed fraction to an improper fraction, dividing fractions by multiplying by the reciprocal, performing multiplication, and converting an improper fraction back to a mixed number. Each of these steps is crucial in mastering fraction operations. Furthermore, we emphasized the importance of understanding number ranges and how to place values within them. By breaking down the problem into manageable steps and explaining the underlying principles, this guide aims to enhance your understanding of fraction arithmetic. Mastering these skills is not only essential for academic success in mathematics but also for various real-world applications. The ability to confidently work with fractions is a valuable asset, and this article provides a solid foundation for further exploration of mathematical concepts. Remember, consistent practice and a clear understanding of the fundamentals are the keys to success in mathematics.

Therefore, the correct answer is:

D) 22 and 23